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HDG methods in finite element exterior calculus

Calvin Reedy, Ari Stern, Yunhui Xue

arXiv:2609.28995v1math.NAcs.NA

TL;DR

FEEC had lacked a unified analysis for equal-order HDG methods, while prior Hodge–Laplace analyses imposed conforming-space or penalty restrictions. This paper develops and analyzes equal-order HDG methods for Hodge–Dirac and Hodge–Laplace problems in arbitrary dimension, proving optimal convergence for Hodge–Dirac components and the Hodge–Laplace solution, with improved auxiliary-variable rates in specified degrees.

  • Problem

    FEEC lacked a unified analysis of equal-order HDG methods, and prior Hodge–Laplace work imposed conforming-complex or penalty restrictions.

    Source: §1 · p. 1

  • Method

    The paper develops and analyzes equal-order HDG methods for Hodge–Dirac and Hodge–Laplace problems in arbitrary dimension using degree-r polynomials and O(1) penalties.

    Source: §1 · p. 2

  • Results

    Hodge–Dirac components converge optimally at order-(r + 1); the Hodge–Laplace k-form solution does too, while auxiliary variables reach that order when k = 1 and/or k = n − 1.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The estimates include lower-regularity cases with reentrant-corner singularities and shape-regular meshes that need not be quasi-uniform.

    Source: §1 · p. 2

    The authors identify arbitrary-topology domains as a direction for future work because Assumptions 4.2 and 5.11 might not hold there.

    Source: §8 · p. 36

Abstract

from arXiv · show

We develop and analyze HDG methods for two central problems in finite element exterior calculus, the Hodge-Dirac problem and the Hodge-Laplace problem, in arbitrary dimension $n$. Our analysis allows for equal-order polynomial spaces with $\mathcal{O}(1)$ penalty parameters, by contrast with previous work on the Hodge-Laplace problem requiring an underlying conforming complex and $\mathcal{O}(h)$ or $\mathcal{O}(h^{-1})$ penalties. For the Hodge-Dirac problem with degree-$r$ polynomials, our error estimates give optimal order-$(r+1)$ convergence in all form degrees under suitable regularity hypotheses. For the Hodge-Laplace problem, we prove optimal order-$(r+1)$ convergence for the $k$-form solution and order-$(r+\frac{1}{2})$ convergence for the $(k \pm 1)$-form auxiliary variables. We obtain improved order-$(r+1)$ auxiliary-variable estimates when $k=1$ and/or $k=n-1$, which in particular sharpens some recent HDG error estimates for the two-dimensional vector Poisson equation. Our analysis also encompasses cases of lower regularity, including solutions with reentrant-corner singularities on non-convex domains, and requires only mild mesh regularity conditions. The results are illustrated by numerical experiments in dimensions two and three.

1. Introduction

The paper extends FEEC’s unified analysis to equal-order HDG methods for the Hodge–Dirac and Hodge–Laplace problems in arbitrary dimension. It establishes convergence results under mild mesh conditions, including lower-regularity cases.

1. Introduction

  • FEEC unifies analyses of scalar and vector finite element methods as differential-form methods, but most prior FEEC analysis concerns conforming methods.Source: §1 · p. 1
  • An open FEEC problem was to analyze equal-order HDG methods; prior XG analysis required a conforming complex and O(h) or O(h^-1) penalties, while the cited 2D HDG estimates had suboptimal auxiliary-variable rates.Source: §1 · pp. 1–2
  • For degree-r polynomials with O(1) penalties, the Hodge–Dirac method achieves optimal order-(r + 1) convergence in all solution components under suitable regularity hypotheses.Source: §1 · p. 2
  • For the Hodge–Laplace problem, the k-form solution converges at order-(r + 1), while auxiliary variables converge at order-(r + 1/2), improving to order-(r + 1) when k = 1 and/or k = n − 1.Source: §1 · p. 2
  • In the 1-form Hodge–Laplace problem in R2, all variables achieve optimal order-(r + 1) convergence, improving on Cockburn et al. [20].Source: §1 · p. 2
  • The analysis includes lower-regularity solutions, including reentrant-corner singularities on non-convex domains, and assumes shape-regular meshes without requiring quasi-uniformity.Source: §1 · p. 2

2. Preliminaries

The preliminaries set up exterior calculus, boundary traces, the Hodge problems, and mesh-skeleton notation used to formulate the HDG methods. They also state the regularity framework and trace conventions needed for the analysis.

2.1. Exterior calculus.

  • The Hodge–Dirac operator D = d + δ mixes form degrees, while its square L = dδ + δd is the Hodge–Laplace operator and preserves degree.Source: §2.1 · p. 3

2.3. The Hodge–Dirac and Hodge–Laplace problems.

  • The Hodge–Dirac problem decomposes f into exact, coexact, and harmonic forms; its well-posedness on Lipschitz domains follows from compact embedding and closed range.Source: §2.3 · p. 4
  • Domain t-regularity gives Sobolev regularity and stability for the Hodge–Dirac and Hodge–Laplace solutions; smooth and convex Lipschitz domains are cited as 1-regular.Source: §2.3 · pp. 4–5
  • The Hodge–Laplace problem seeks u and p, with auxiliary forms σ = δu and ρ = du, and yields the decomposition f = dσ + δρ + p.Source: §2.3 · p. 5

2.4. Broken spaces, jumps, and averages.

  • On the mesh skeleton, traces are generally double-valued; conforming spaces require single-valued tangential or normal traces, with orientation-dependent sign conventions.Source: §2.4 · pp. 5–6
  • Jumps and averages are orthogonal projections whose sum recovers each trace; the chosen boundary convention suits problems with normal boundary conditions.Source: §2.4 · p. 6

3. HDG methods for the Hodge–Dirac problem

This section defines an equal-order HDG weak formulation for the Hodge–Dirac problem and establishes its solvability under an alternating-sign penalty condition. It also gives an equivalent symmetric jump-average DG formulation.

3.1. Weak formulation.

  • Integration by parts yields the weak formulation, which uses equal-order degree-r polynomial spaces.The formulation also includes harmonic spaces.Source: §3.1 · p. 7
  • Local harmonics do not affect the other solution components, but they are needed for well-defined local solvers and static condensation.Static condensation eliminates local variables to obtain a smaller system in the global variables.Source: §3.1 · p. 8, §3.2 · p. 8
  • The method is a special case of the LDG-H methods in Stern and Zampa [36], while its treatment of harmonics, boundary conditions, and analysis is new.The cited work leaves open the treatment of harmonic forms and boundary conditions needed to specify a particular solution.Source: §3.1 · p. 8

3.2. Existence and uniqueness.

  • Under the alternating-sign penalty assumption, the method has unique local solutions and a unique global solution.The assumption requires (−1)^k α_k to be strictly positive for all k or strictly negative for all k.Source: §3.2 · p. 9

3.3. Jump-average DG formulation.

  • Eliminating the hybrid variables gives an equivalent jump-average DG formulation, which is symmetric.The equivalent formulation is used for the subsequent error analysis.Source: §3.3 · pp. 10–11

4. Error analysis for the Hodge–Dirac problem

This section develops Hodge–Dirac error estimates, combining abstract bounds with approximation estimates to obtain solution and trace estimates under regularity and mesh assumptions. It also establishes improved 1-form error estimates in two dimensions under Theorem 4.10’s hypotheses.

4.1. Bilinear forms.

  • A sign choice based on the opposite signs of α+ and α− converts the symmetric indefinite form into a nonsymmetric positive-semidefinite form with an associated seminorm.If the initial sign choice yields a negative-semidefinite form, flipping its sign makes it positive-semidefinite.Source: §4.1 · p. 12

4.2. Abstract error estimates.

  • The A-seminorm error is bounded by KA(z, z) + KB(z)^2, while an adjoint duality argument yields an L2 error estimate.These abstract estimates assume H(Ω) ⊂ Wh, which gives Galerkin orthogonality and places the error in H(Ω)⊥.Source: §4.2 · pp. 13–14

4.3. Approximation theory: determining KA and KB.

  • The approximation bounds for KA and KB hold on meshes with affine-equivalent cells, comparable neighboring diameters, and shape regularity, using O(1) penalty parameters.The mesh assumptions admit varied cell types, including curved elements, and hanging nodes.Source: §4.3 · pp. 14–15

4.4. Final error estimates.

  • Under Ω t-regularity and z ∈ H^sΛ(Ω), with 1/2 < t ≤ s ≤ r + 1, Theorem 4.10 gives degree-r HDG error estimates.Under the same hypotheses, Theorem 4.12 also estimates the approximate traces.Source: §4.4 · pp. 16–17
  • For minimum regularity, the worst-case convergence rates remain positive but approach zero as t approaches 1.This scenario assumes Ω is t-regular for 1/2 < t ≤ 1 and z ∈ H^tΛ(Ω).Source: §4.4 · p. 18

4.5. Improved estimates in dimension two.

  • In dimension two, Theorem 4.14 gives improved error estimates for the 1-form solution and its approximate traces under Theorem 4.10’s hypotheses.The improvement is available even if the domain fails to be 1-regular at other degrees.Source: §4.5 · p. 18
  • The proof uses 1-form duality, whose adjoint solution has only 0- and 2-form components, and treats jumps and averages separately in the trace estimates.This avoids potentially suboptimal L2-norm estimates for e0 z.Source: §4.5 · pp. 18–19

5. HDG methods and error analysis for the Hodge–Laplace problem

This section adapts the HDG framework to the Hodge–Laplace problem and establishes well-posedness and error estimates with equal-order spaces and O(1) penalties. It analyzes convergence rates for the solution, auxiliary variables, and traces, including improved rates for specific form degrees and lower regularity; in the two-dimensional 1-form problem, all fields converge optimally.

5.1. Weak formulation.

  • The method uses the same equal-order finite element spaces as the Hodge–Dirac problem, but requires forms only in degrees k−1, k, and k+1 rather than every degree.An alternative hybridization requires no local harmonics and yields a symmetric positive-definite Schur complement modulo the nullspace of global harmonics; previous hybridizations generally yield saddle-point problems.Source: §5.1 · pp. 19, 21

5.2. Existence and uniqueness.

  • Under αk−1 > 0 and αk < 0, the HDG system has a unique solution for every f ∈ L2Λk(Ω).The analysis establishes local existence and uniqueness for a prescribed uh before proving global existence and uniqueness.Source: §5.2 · p. 21

5.3. Jump-average DG formulation.

  • The approximate traces also admit a jump-average DG formulation, which the error analysis uses in unhybridized form.The trace relations are stated to hold when the penalty coefficients αk−1 and αk are single-valued.Source: §5.3 · pp. 22–23

5.4.1. Bilinear forms.

  • The Hodge–Laplace bilinear forms reuse the Hodge–Dirac analysis but omit terms associated with (k ± 2)-forms, which are not involved in this problem.A modified seminorm equivalence suffices because the omitted jump terms do not occur in the Hodge–Laplace problem.Source: §5.4.1 · pp. 23–24

5.4.2. Abstract error estimates.

  • The abstract estimates use a degree-k harmonic-form assumption and an adjoint problem with nonzero λk±1 components to obtain duality estimates for auxiliary-variable errors.The resulting estimates combine A- and C-seminorm bounds with an L2 error estimate.Source: §5.4.2 · pp. 24–25

5.4.3. Final error estimates.

  • For the degree-r HDG solution of the k-form Hodge–Laplace problem, Theorem 5.14 establishes error estimates when Ω is t-regular, z ∈ H^sΛ(Ω), 1/2 < t ≤ s ≤ r+1, with mesh regularity and O(1) penalties.Source: §5.4.3 · p. 26
  • Improved auxiliary-variable estimates apply when k = 1 and/or k = n − 1; in general, only their projections are guaranteed the improved rate.The auxiliary-variable L2 errors are generally controlled by |z − zh|C, while the projections P˚ B∗σh and PBρh have the improved rate.Source: §5.4.3 · p. 26
  • O(r+1) convergence holds for all fields in the two-dimensional 1-form Hodge–Laplace problem, sharpening the estimates of Cockburn et al. [20] for the vector Laplacian.For minimum regularity, the same problem has order 2t − 1 convergence when 1/2 < t ≤ 1.Source: §5.4.3 · pp. 27–28
  • O(r+1) convergence holds for the solution and its approximate traces; auxiliary variables and traces converge at order r+1/2 or r+1, depending on k.These are the best-case rates when regularity allows.Source: §5.4.3 · p. 28

6. An alternative hybridization for the Hodge–Laplace problem

The alternative Hodge–Laplace hybridization uses normal traces as global variables and yields the same numerical solutions as the earlier formulation. It avoids local harmonic forms and supports static condensation with a symmetric positive-definite Schur complement modulo global harmonics.

6. An alternative hybridization for the Hodge–Laplace problem

  • Using normal traces as global variables gives the same numerical solutions as the earlier hybridization while avoiding local harmonic forms.The authors present these as possible implementation advantages.Source: §6 · p. 28, §6.1 · p. 29

6.1. Weak formulation.

  • The alternative weak formulation has local existence and uniqueness, and is equivalent to the previous formulation.Its solutions therefore coincide with those of the earlier formulation.Source: §6.1 · p. 29

6.2. Static condensation.

  • Static condensation gives a symmetric positive-definite bilinear form on the trace space modulo global harmonics, yielding a unique equivalence class of solutions.When global harmonics are trivial, the condensed system is positive-definite without taking the quotient.Source: §6.2 · pp. 30–31

7. Numerical experiments

The numerical experiments in two and three dimensions illustrate the equal-order HDG methods’ convergence behavior for the Hodge–Dirac and Hodge–Laplace problems. On smooth domains, observed rates match the estimates; on the L-shaped domain, the experiments show the predicted lower-regularity rates.

7.1. Hodge–Dirac.

  • On smooth unit-square and unit-cube tests, Hodge–Dirac errors converge at order r + 1/2 in the A seminorm and order r + 1 in L2, matching Theorem 4.10.The square and cube experiments use smooth solutions on convex domains.Source: §7.1 · pp. 32, 34, Table 1 · p. 32, Table 3 · p. 33
  • For the minimum-regularity Hodge–Dirac problem on the L-shaped domain, observed rates match the estimates, including an improved L2-norm convergence rate of 2/3 for z1.Source: §7.1 · p. 32, Table 2 · p. 33

7.2. Hodge–Laplace.

  • On the smooth unit-square 1-form Hodge–Laplace test, all three components show the expected rates: r + 1/2 in the A seminorm and r + 1 in L2.Here σ = −div u and ρ = curl u.Source: Table 4 · p. 34, §7.2 · p. 35
  • On the minimum-regularity L-shaped 1-form Hodge–Laplace test, all components show order 1/6 in the A seminorm and order 1/3 in L2; u has no improved L2 rate.These rates match Theorem 5.14 and Corollary 5.15.Source: §7.2 · p. 35, Table 5 · p. 35
  • For the smooth 2-form Hodge–Laplace test, the unit-cube experiment shows order r + 1 convergence for u and ρ in L2, and also for σ.The σ rate exceeds its order-(r + 1/2) estimate; k = n − 1 in this test.Source: §7.2 · p. 36, Table 6 · p. 36

8. Concluding remarks

The conclusion identifies open directions in extending the analysis to arbitrary topology, sharpening auxiliary-variable convergence rates, and developing efficient solvers.

8. Concluding remarks

  • Extending the methods to domains with arbitrary topology may require replacing Hh with discrete harmonic forms and controlling the resulting variational crime.Such discrete harmonics are generally not contained in H(Ω), so additional analysis such as inf-sup stability might be needed.Source: §8 · p. 36
  • For general k, it remains open whether auxiliary-variable convergence can improve from order-(r + 1/2) to order-(r + 1), beyond the cases k = 1 and k = n −1.Optimal-order estimates are known for k = 0 and k = n in the scalar Poisson equation; extending HDG projection and M-decomposition techniques is proposed.Source: §8 · p. 36
  • Efficient solvers and preconditioners remain to be investigated, particularly for the alternative Hodge–Laplace hybridization with a symmetric positive-(semi)definite condensed system.The authors suggest that this system might be solved by conjugate gradients with an appropriately devised preconditioner.Source: §8 · p. 36
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